Variable Speed Limit Sign (VSLS) Systems enable speed limits to be changed dynamically in response to traffic conditions so that traffic incidents can be reduced significantly on freeway work zones. In this paper, we examined how many and where VSLS are to be placed and the speed limits to be set and proposed a bilevel programming model to perform this decision making operation. The appropriate speed limits and deployments of VSLS were got by case study, and they were analyzed by a simulation to prove the empirical features of traffic breakdown at freeway work zones. Then the results of model comparison and simulation evaluation illustrate that the proposed method outperforms existing models in terms of maximizing information benefit and minimizing average queue length, total delay, and total stop frequency on the freeway work zone.
Uneven velocity distribution is a major reason that contributes to incidents (e.g., traffic accident or congestion) in freeway work zones. In freeway work zones, the speed of vehicles will become discrete from upstream section to downstream section because of the temporarily closed lane, which will cause a series of traffic behaviors like lanechanging, slowing down suddenly, and carflowing. So once the incident happens, these behaviors may lead to a secondary accident or a sharp decline of traffic capacity of freeway work zones [
“Variable Speed Limit Sign (VSLS) Control” is an effective traffic control technique that has been used in freeway work zones for recent years [
There has been considerable research work on the first aspect. A number of scholars (e.g., Mcmurtry et al. [
The above literatures have studied the speed limits or deployments of VSLS independently, while these theoretical methods are still the classical traffic theories which have been criticized in some latest reviews [
In this paper, the main idea is to explore the comprehensive control effect of speed limit values and deployments of VSLS. On the basis of the above studies, we attempt to propose a bilevel programming model to seek for the best suitable speed limits and the corresponding locations of VSLS. The first object of the bilevel programming model is to optimize the numbers and speed limit values of VSLS by establishing a minimum comprehensive accident rate model. The second object is to optimize the locations of VSLS by solving the improved maximum information benefit model.
The remainder of this paper is organized as follows: In Section
This section first introduces the freeway work zone and then describes model development, that is, the bilevel programming model and its two objective functions. The first objective function is a comprehensive accident rate model, and its influencing factors are the numbers and speed limit values of VSLS; that is, the comprehensive accident rate will be different according to various numbers of VSLS and its speed limit values. Our goal is to find the appropriate numbers and speed limit values of VSLS which can minimize the comprehensive accident rate. The second objective function is a maximum information benefit model in which the different locations of VSLS are influencing factors; that is, the best appropriate locations of VSLS should correspond to the largest information benefit. After that, the model comparison and simulation evaluation will be proposed in Section
According to “Freeway Maintenance Safety Operating Procedures (JTGH302004)” of China, freeway work zones contain six parts: warning zone, upstream transition zone, buffer, workspace, downstream transition zone, and termination zone, as shown in Figure
Composition of freeway work zones.
Assuming that an incident happens at the upstream transition zone at a certain time, then the volume and speed collected by detectors are transmitted to the traffic control center and then the relative control models, that is, the two objective functions of the bilevel programming model, are applied to determine the VSLS numbers, the speed limits, and VSLS locations. And the speed limit values can be adjusted dynamically, and then they will be allocated to the corresponding VSLS, and the numbers and locations of VSLS can be also adjusted dynamically.
Traffic flow at freeway work zones has a very complicated feature, the classical traffic flow theory includes two phases, that is, smoothing and blocking flow, which cannot demonstrate the actual characteristics of traffic flow perfectly at freeway work zones sometimes. So the “threephase traffic theory” was proposed by Kerner [
Threephase traffic theory model of bottlenecks.
This section analyzes the actual traffic characteristics of freeway work zones from temporalspatial perspective. The traffic volume is selected from the freeway of Shandong, China, the detected sections include K12+875, K14+795, K16+985, and detected time is from October 20, 2014, to October 26, 2014. The spatial characteristics are divided into two aspects, one is lateral aspect (the adjacent lanes, e.g., lane 1 and lane 2 of K12+875), and the other is the longitudinal aspect (the adjacent sections, e.g., the upstream section (K12+875), the middle section (K14+795), and the downstream section (K16+985)). Temporalspatial characteristics of traffic flow and speed at freeway work zone of Shandong are shown in Figures
Temporalspatial characteristics of traffic flow.
Traffic characteristics of longitudinal section
Traffic characteristics of lateral section
Speed characteristics of lateral section.
Figures
Traffic accidents happen frequently at freeway work zones. In general, the accident rates of freeway work zones in northeast China are 2.7 times as high as that in ordinary road [
Pu et al. have studied the relationship between vehicle speed and accident rate by analyzing the actual freeway data [
Zhong et al. have studied the relationship between interval speed difference and accident rate [
Furthermore, Hou et al. have studied the relationship between speed standard deviation and accident rate [
The above models considered single factor only (e.g., speed, speed difference, or speed standard deviation) in their respective study. By analyzing (
In order to explore problems further, we presume that all the vehicles drive at the speed of VSLS at the control zone. As a result of the above analysis, a minimum comprehensive accident rate model is established as follows:
In this section, we first introduce the information benefit; several literatures have introduced the information benefit (e.g., Ni and Liu, 2003 [
The division of VSLS control zone.
The first maximum information benefit model (i.e., the traditional model) was proposed by Ni and Liu to explore the deployment of VMS (in this paper it is also applied to VSLS), the model shown as follows:
Then the principle of the maximum information benefit model is as follows.
At freeway work zones, drivers can pay more attention to speed limit information from the effective deployments of VSLS. In other words, drivers can obtain more information benefits from the reasonable setting of VSLS. Thus we use the idea of “maximum information benefit” to study the VSLS locations; the largest information benefits in the objective function obviously correspond to the best appropriate locations of VSLS.
Now the improved maximum information benefit model is introduced, in which the main improvements influence index and attenuation coefficient. Firstly, in Section
Secondly, we improved the attenuation coefficient. In traditional model, the attenuation coefficient only depends on the quantity of road from the current VSLS location to the incident site. Actually it is also affected by travel time of each VSLS control area which has been proposed in some papers. Hence, the improved attenuation coefficient should be
Finally, the improved maximum information benefit model can be expressed as follows:
The bilevel programming model in this paper includes the following two objects: the first object is to optimize the speed limits and numbers of VSLS, and the second object is to optimize the locations of VSLS. The bilevel programming model is
In this section, the corresponding algorithms are performed with the experiment. The two objective functions of the bilevel programming model are solved by different algorithms. The first objective function is solved by exhaustive algorithm, and the second objective function is solved by genetic algorithm.
Generally, the more the VSLS, the better the control effects. However, setting too many VSLS can not only waste resources but also lose drivers’ trust. Moreover, it can lead to other safety problems because of the frequent distractions. Therefore, the VSLS setting numbers of freeway work zones should be moderate. And the cost factor should be considered.
An exhaustive method is used to solve this objective function. Firstly, we assume the initial speed
Result of minimum comprehensive accident rate model.




1  2.247  100 
2  1.124  100, 90 
3  0.749  100, 90, 80 
4  0.562  100, 90, 80, 70 
5  0.450  100, 90, 80, 70, 60 
6  0.375  100, 90, 80, 70, 60, 50 
7  0.321  100, 90, 80, 70, 60, 50, 40 
8  0.281  100, 90, 80, 70, 60, 50, 40, 30 
As can be seen from Table
Reduction trend of the minimum comprehensive accident rate.
However, in reality, vehicles could slow down in advance due to the warning sign, so
Actual speed limits of the four VSLS.





50 km/h  40 km/h  30 km/h  20 km/h 
As shown in Table
Now we consider that the speed distribution at freeway work zones is from 50 km/h to 20 km/h. The speed limits
Relationship between flow rate and vehicle density.
Temporalspatial distribution of traffic flow at freeway work zones.
In this section, genetic algorithm is applied to solve the second objective function. This is because the second objective function can be treated as a mathematical optimization problem, and the major optimal objects are VSLS locations. In the design of genetic algorithm, fitness represents the objective function value (i.e., the information benefit), so if fitness is bigger, information benefit is also bigger; then the location of VSLS is better. Genetic algorithm includes the following steps.
From Section
VSLS distribution of each control area.
A simulated road network of the freeway work zone is built to show the deployments of the four VSLS (see Figure
8 locations are chosen as the alternative locations of VSLS, because if they are beyond 8, the calculation is very complicated, while if they are too small, the convergence will be too fast, so after many trials, the alternative locations are 8.
The assumption is that an incident happens at the upstream transition zone at a certain time, because only in this condition would the study be meaningful.
Each road length is 500 m with a traffic volume of 500 pcu/h when the incident happens (this hypothesis is convenient for our simulation with VISSIM).
Simulation network of freeway work zone.
Now we need to set four VSLS to get more information benefits, and under this condition, the traffic situation and the secondary accident rate could be decreased. The speed limits of the four VSLS are
In this section, four models are used for comparison; apart from the improved information benefit model (the second objective function of the bilevel programming model) and traditional information benefit model that Ni and Liu have proposed, another two models are proposed by this paper so as to make a more comprehensive contrast. In this paper, model 4 (traditional model which is the existing model) is the criterion for the comparison, and the other three models are all improved on the basis of model 4.
The four models are expressed as follows:
Model 1: improved maximum information benefit model, that is, the second objective function of the bilevel programming model
Model 2: maximum information benefit model where the attenuation coefficient is constant (it is the same as traditional information benefit model) and the influence index is the same as model 1
Model 3: maximum information benefit model where the influence index is constant (it is the same as traditional information benefit model) and the attenuation coefficient is the same as model 1
Model 4: traditional information benefit model proposed by Ni and Liu.
Genetic algorithm is used to solve these models, and the results are
Results comparison of the four models.
Model 1
Model 2
Model 3
Model 4
From the largest information benefits of four models, it can be seen that the best deployment of VSLS comes from model 1, followed by model 2, model 3, and model 4. Their deployments are in locations 1, 2, 3, and 8; 1, 2, 3, and 4; 1, 2, 3, and 8; 4, 5, 6, and 7, respectively. Figure
In terms of different deployments of VSLS and their corresponding speed limits at freeway work zones, the situation of traffic flow of each deployment (i.e., locations 1, 2, 3, and 8; locations 1, 2, 3, and 4; locations 4, 5, 6, and 7) is simulated, respectively, by VISSIM. Two simulations are performed; the first is to show if the relative deployments of VSLS at freeway work zones are consistent with the actual traffic situation of freeway work zones. Because the velocity distribution of each deployment (i.e., locations 1, 2, 3, and 4; locations 1, 2, 3, and 8; and locations 4, 5, 6, and 7) is 50 km/h, 40 km/h, 30 km/h, and 20 km/h, we use this velocity distribution to simulate the actual traffic flow. In Figures
Speed distribution of different deployments.
Speed distribution of locations 1, 2, 3, and 4
Speed distribution of locations 1, 2, 3, and 8
Speed distribution of locations 4, 5, 6, and 7
The three different deployments of VSLS in Figure
The second simulation is to evaluate the relevant parameters (e.g., average queue length, travel time, total stop frequency, and total delay) of each model to show the superiority of the model proposed in this paper (model 1, improved maximum information benefit model). Total simulation time is 4500 seconds and every 20 seconds a count is made. Total length of simulation road is the same as the road in Figure
Evaluation results of different deployments.
Average queue length of different deployments
Travel time of different deployments
Stop frequency of locations 1, 2, 3, and 4
Stop frequency of locations 1, 2, 3, and 8
Stop frequency of locations 4, 5, 6, and 7
Traffic delay of different deployments
Figures
All of the above indicate that the effective VSLS control can not only be consistent with the empirical nucleation nature of traffic breakdown but also reduce average queue length, total delay, and total stop frequency.
A bilevel programming model is established to study the VSLS control of freeway work zones. The first object of the bilevel programming model is to minimize comprehensive accident rate, where VSLS numbers and the corresponding speed limits are determined. The second object is to maximize the information benefit, where the best deployment of VSLS is obtained. Case study shows the following results:
The best appropriate number of VSLS at the freeway work zone of the simulated road network is 4 and the corresponding speeds are 50 km/h, 40 km/h, 30 km/h, and 20 km/h, respectively.
The best deployments of VSLS are locations 1, 2, 3, and 8 at the freeway work zone of the simulated road network.
Results of our method (the two objective functions) show that the traffic flow is consistent with “threephase theory” at the known conditions; that is,
Despite the fact that implementing effective VSLS control can reduce the comprehensive accident rate and smoothen the traffic flow of the freeway work zone, there are still several shortcomings in the study. For instance, the types of the different work zones are not fully considered, the empirical features of traffic breakdown at freeway work zones need to be further studied, and some assumptions (e.g., the initial speed
The authors declare that they have no conflicts of interest.
This research has been jointly supported by National Key Technology Support Program (Grant no. 2014BAG03B03).